These expressions are physically intuitive and useful since all the quantities involved are those in 3D. These quantities are well known and most of them are tightly constrained either theoretically or experimentally. Let's now discuss a few main features if a nonzero \(\epsilon \)  is applied. Besides the kinetic contribution, a non-trivial contribution to the EOS of SNM comes from the integration over the momentum-dependence of the potential \(U_0\) , i.e., \((\sigma /\rho )\int _0^{\rho }[\overline{k}_{\mathrm {F}}^f\cdot \partial U_0/\partial |\mathbf {k}|]_{|\mathbf {k}|=\overline{k}^f_{\mathrm {F}}}\mathrm {d}f\) . One knows from nuclear optical model fitting to nucleon-nucleus scattering data that the dependence of the nucleon isoscalar potential \(U_0\)  on the nucleon momentum is positive [1]}, [2]}, [3]}, thus the integration is also positive since \(\sigma >0\) . Consequently the linear \(\epsilon \) -term contributes a positive term to the unperturbed EOS of SNM, as demonstrated in formula (REF ).
If we extrapolate the 3D EOS of SNM to lower dimensions (higher dimensions), it will be reduced (enhanced). Specifically, the 2D SNM is much more bounded than its 3D counterpart.
For the symmetry energy, we first neglect the momentum-dependence of the symmetry potential as well as the second derivative \(\partial ^2U_0/\partial |\mathbf {k}|^2\) , then the \(\epsilon \) -term of () becomes
\(\epsilon \left[\frac{\overline{k}_{\mathrm {F}}^2}{6M}\left(2\sigma -\frac{1}{3}\right)+\frac{\overline{k}_{\mathrm {F}}}{6}\frac{\partial U_0}{\partial |\mathbf {k}|}\left(\sigma -\frac{1}{3}\right)_{|\mathbf {k}|=\overline{k}_{\mathrm {F}}}\right]=\frac{\overline{k}_{\mathrm {F}}^2}{6M}\left[\frac{M}{M_{\mathrm {s}}^{\ast }}\left(\sigma -\frac{1}{3}\right)+\sigma \right]\epsilon \approx \frac{\overline{k}_{\mathrm {F}}^2}{6M}\left(0.24-0.09\frac{M}{M_{\mathrm {s}}^{\ast }}\right)\epsilon ,\) 
